Thread: Anti-derivatives - Proof?

Greetings all,

We recently finished up a section on area and volume using integrals and we were learning about what the anti-derivative is. My question is I am not sure if there is proof that the anti-derivative can give us the area or volume. I asked my teacher about this but there doesn't seem to be a concrete proof? I am just wondering how did Newton or what not come to learn that the anti-derivative, which I see as the pre-slope leads to the area and volume?

Right now I am a skeptic about the anti derivative. Can someone make me a believer? I need some proof. I am just not convinced.

The connection between antiderivatives and definite integrals is given by the fundamental theorem of calculus.

Thanks for the link

we talked about the fundamental theorem of calculus. maybe I'm needing something that explains it better. are there are any videos? I've tried khan academy but couldn't find it. I will go over my notes but I'm looking for a more simpler answer like the overall concept, and then maybe I'll be able to understand the proof better. I'll probably print up the wiki page and go over it in the aid center at my school. I find going over proof difficult but I am interested in getting to the heart of the antiderivative.

An outline of the fundamental theorem of calculus is given in this link from this thread. The link says that it deals with the second theorem, but it seems to me that it is a proof outline of the first fundamental theorem of calculus: assuming F is a definite integral of f, it shows that F' = f. It is not a complete proof because it uses the approximate equality sign. However, it gives the idea of the proof, and it can be made precise. The Wiki page for the fundamental theorem of calculus presumably contains a full proof (I have not studied it).

Originally Posted by funnybabe

I find going over proof difficult but I am interested in getting to the heart of the antiderivative.

That's the right spirit.